class: center, middle, inverse, title-slide # Lecture 12 ## Models for Factorial Designs ### Psych 10 C ### University of California, Irvine ### 04/25/2022 --- ## Models for factorial designs - The previous class we talked about two different models for factorial designs. -- - The **Null** model which formalizes the assumption that the combinations of our factors (groups) have no effect on the expectation of our dependent variable (observations). -- - The Null model is expressed formally as: `$$y_{ijk}\sim\text{Normal}(\mu,\sigma_0^2)$$` - Where `\(i\)` represents the observation number of the combination of the *j-th* level of factor 1 and the *k-th* level of factor 2. -- - The second type of model we covered was the **Main effects** model. Main effects models assume that the expected value of our dependent variable is different between levels of a single factor regardless of the values of other factors on the experiment. -- - The number of Main effects models that we will have depends on the number of independent factors that we have. --- ## Models for factorial designs - As we saw last class, a Main effects model for factor `\(j\)` was expressed as: `$$y_{ijk}\sim\text{Normal}(\mu+\alpha_j,\sigma_1^2)$$` -- - While the main effects model of factor `\(k\)` is: `$$y_{ijk}\sim\text{Normal}(\mu+\beta_k,\sigma_2^2)$$` -- - We will only work with `\(2\times 2\)` factorial designs for now, so these are the only two main effects models that we need. -- - Remember that we use a different effects variable `\((\alpha_j\)` for factor `\(j\)` and `\(\beta_k\)` for factor `\(k\)`) because we will use those variables for another model. -- - Today, we will introduce the remaining 2 models that we use in a factorial design and work on an example using data from our anxiety example with different cohorts of students that took either a statistics course during the year or another class.